Key Features
- Proof of Bieberbach conjecture (after DeBranges)- Material on asymptotic values- Material on Natural Boundaries- First four chapters are comprehensive introduction to entire and metomorphic functions- First chapter (Riemann Mapping Theorem) takes up where "first courses" usually leave offForeword
A Note on Notational Conventions
Chapter 1: Conformal Mapping and the Riemann Mapping Theorem
1.1 Introduction
1.2 Linear fractional transformations
1.3 Univalent Functions
1.4 Normal Families
1.5 The Riemann Mapping Theorem
Chapter 2: Picard’s Theorems
2.1 Introduction
2.2 The Bloch-Landau Approach
2.3 The Elliptic Modular Function
2.4 Introduction
2.5 The Constants of Bloch and Landau
Chapter 3: An Introduction to Entire Functions
3.1 Growth, Order, and Zeros
3.2 Growth, Coefficients, and Type
3.3 The Phragmén-Lindelöf Indicator
3.4 Composition of entire functions
Chapter 4: Introduction to Meromorphic Functions
4.1 Nevanlinna’s Characteristic and its Elementary Properties
4.2 Nevanlinna’s Second Fundamental Theorem
4.3 Nevanlinna’s Second Fundamental Theorem: Some Applications
Chapter 5: Asymptotic Values
5.1 Julia’s Theorem
5.2 The Denjoy-Carleman-Ahlfors Theorem
Chapter 6: Natural Boundaries
6.1 Natural Boundaries—Some Examples
6.2 The Hadamard Gap Theorem and Over-convergence
6.3 The Hadamard Multiplication Theorem
6.4 The Fabry Gap Theorem
6.5 The Pólya-Carlson Theorem
Chapter 7: The Bieberbach Conjecture
7.1 Elementary Area and Distortion Theorems
7.2 Some Coefficient Theorems
Chapter 8: Elliptic Functions
8.1 Elementary properties
8.2 Weierstrass’ -function
8.3 Weierstrass’ ?- and s-functions
8.4 Jacobi’s Elliptic Functions
8.5 Theta Functions
8.6 Modular functions
Chapter 9: Introduction to the Riemann Zeta-Function
9.1 Prime Numbers and ?(s)
9.2 Ordinary Dirichlet Series
9.3 The Functional Equation, the Prime Number Theorem, and De La Vallée-Poussin’s Estimate
9.4 The Riemann Hypothesis
Appendix
1 The Area Theorem
2 The Borel-Carathéodory Lemma
3 The Schwarz Reflection Principle
4 A Special Case of the Osgood-Carathéodory Theorem
5 Farey Series
6 The Hadamard Three Circles Theorem
7 The Poisson Integral Formula
8 Bernoulli Numbers
9 The Poisson Summation Formula
10 The Fourier Integral Theorem
11 Carathéodory Convergence
Bibliography
Index
